A straightforward extension of the signal detection theory (SDT) framework is described and demonstrated for the two-alternative identification task. The extended framework assumes that the subject and an arbitrary model (or two subjects, or the same subject on two occasions) are performing the same task with the same stimuli, and that on each trial they both compute (in effect) values of a decision variable. Thus, their joint performance is described by six fundamental quantities: two levels of intrinsic discriminability ("d-prime"), two values of decision criterion, and two decision-variable correlations, one for each of the two categories of stimuli in the task. Decision-variable correlations (DVCs) provide increased power for testing models and for characterizing individual differences, and do not require a special experimental design (e.g., they can be computed from existing data). The extended framework was developed for analyzing trial-by-trial performance in vision experiments with natural stimuli, but it should be widely applicable in behavioral and neurophysiological studies of perception and cognition. We demonstrate the framework for the well-known task of detecting a Gaussian target in white noise and make several theoretical and experimental discoveries: (1) subjects' DVCs are approximately equal to the square root of their efficiency relative to ideal (in agreement with the prediction of a popular class of models), (2) between-subject and within-subject (double-pass) DVCs increase with target contrast and are greater for target-present than target-absent trials (rejecting many models), (3) model parameters can be estimated by maximizing DVCs between the model and subject, (4) a model with a center-surround template and a specific (modest) level of position uncertainty predicts the trial-by-trial performance of subjects as well (or better) than presenting the same stimulus again to the subjects (i.e., the double-pass DVCs, which are as high as 0.7). We conclude that measuring DVCs can be of considerable value.